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 A070003 Numbers divisible by the square of their largest prime factor. 42
 4, 8, 9, 16, 18, 25, 27, 32, 36, 49, 50, 54, 64, 72, 75, 81, 98, 100, 108, 121, 125, 128, 144, 147, 150, 162, 169, 196, 200, 216, 225, 242, 243, 245, 250, 256, 288, 289, 294, 300, 324, 338, 343, 361, 363, 375, 392, 400, 432, 441, 450, 484, 486, 490, 500, 507 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Numbers n such that P(phi(n)) - phi(P(n)) = 1, where P(x) is the largest prime factor of x. P(phi(n)) - phi(P(n)) = A006530(A000010(n)) - A000010(A006530(n)). Numbers n such that the value of the commutator of phi and P functions at n is -1. Equivalently, n such that n and phi(n) have the same largest prime factor since Phi(p) = p-1 if p is prime. - Benoit Cloitre, Jun 08 2002 Since n is divisible by P(n)^2, n cannot divide P(n)! and so A057109 is a supersequence. Hence all A002034(a(n)) are composite. - Jonathan Sondow, Dec 28 2004 A225546 defines a self-inverse bijection between this sequence and A335740, considered as sets. - Peter Munn, Jul 19 2020 LINKS Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 Paul Erdős and Ron L. Graham, On products of factorials, Bull. Inst. Math. Acad. Sinica 4:2 (1976), pp. 337-355. [alternate link] Paul Erdős and Ilias Kastanas, Solution 6674:The smallest factorial that is a multiple of n, Amer. Math. Monthly 101 (1994) 179. A. J. Kempner, Miscellanea, Amer. Math. Monthly, 25 (1918), 201-210. See Section II, "Concerning the smallest integer m! divisible by a given integer n." Eric Weisstein's World of Mathematics, Greatest Prime Factor Eric Weisstein's World of Mathematics, Totient Function FORMULA Erdős proved that there are x * exp(-(1 + o(1))sqrt(log x log log x)) members of this sequence up to x. - Charles R Greathouse IV, Mar 26 2012 MATHEMATICA p[n_] := FactorInteger[n][[-1, 1]]; ep[n_] := EulerPhi[n]; fQ[n_] := p[ep[n]] == 1 + ep[p[n]]; Select[ Range[ 510], fQ] (* Robert G. Wilson v, Mar 26 2012 *) Select[Range, FactorInteger[#][[-1, 2]] > 1 &] (* T. D. Noe, Dec 06 2012 *) PROG (PARI) for(n=3, 1000, if(component(component(factor(n), 1), omega(n))==component(component(factor(eulerphi(n)), 1), omega(eulerphi(n))), print1(n, ", "))) (PARI) is(n)=my(f=factor(n)[, 2]); f[#f]>1 \\ Charles R Greathouse IV, Mar 21 2012 (PARI) sm(lim, mx)=if(mx==2, return(vector(log(lim+.5)\log(2)+1, i, 1<<(i-1)))); my(v=); forprime(p=2, min(mx, lim), v=concat(v, p*sm(lim\p, p))); vecsort(v) list(lim)=my(v=[]); forprime(p=2, sqrt(lim), v=concat(v, p^2*sm(lim\p^2, p))); vecsort(v) \\ Charles R Greathouse IV, Mar 27 2012 (Python) from sympy import factorint def ok(n): f = factorint(n); return f[max(f)] >= 2 print(list(filter(ok, range(4, 508)))) # Michael S. Branicky, Apr 08 2021 CROSSREFS Subsequence of A057109, A122145. Complement within A020725 of A102750. Cf. A000010, A006530, A068211, A070777, A070812, A070002, A070004, A007283, A070813, A070814, A070815, A070816, A002034, A102067, A102068. Related to A335740 via A225546. A195212 is a subsequence. Cf. A319988 (characteristic function). Positions of odd terms > 1 in A122111. Sequence in context: A140269 A339744 A226385 * A325661 A073539 A090779 Adjacent sequences:  A070000 A070001 A070002 * A070004 A070005 A070006 KEYWORD nonn AUTHOR Labos Elemer, May 07 2002 EXTENSIONS New name from Jonathan Sondow and Charles R Greathouse IV, Mar 27 2012 STATUS approved

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Last modified September 27 10:28 EDT 2021. Contains 347689 sequences. (Running on oeis4.)